Non-triviality of the A-polynomial for knots in S³
The A-polynomial of a knot in S³ defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL₂C. Here, we show that a non-trivial knot in S³ has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU₂-representations of Dehn surgeries on knots in S³. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot.
Additional InformationSubmitted: 13 June 2004. Accepted: 16 September 2004. Published: 1 December 2004. Both authors were partially supported by the U.S. National Science Foundation, and Dunfield was also partially supported by the Sloan Foundation.
Published - DUNagt04.pdf
Submitted - 0405353v1.pdf